# Finding a basis for subspaces of the complex vector space.

Each of the following sets spans a subspace of the complex vector space of all functions from $\mathbb C$ to $\mathbb C$. In each case find a basis for the subspace and prove it is a basis; state the dimension

{$\exp(iz); \cos z;\sin z$}

{$\exp(iz); \cosh z;\sinh z$}

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Please don't just dump undigested problems here. Tell us what you know about the problem, what steps you have taken to solve it, where you get stuck, and so on, so we can give more useful advice. – Gerry Myerson Dec 13 '12 at 1:49

## 1 Answer

Hint for your first case $\;\{e^{iz}, \cos z, \sin z\}\;$:

Recall that $\;e^{iz} = \cos(z) + i\sin(z)$: this is a linear combination of $\cos z$ and $\sin z$.

What does this tell you about the linear-(in)dependence of the the vectors $e^{iz}, \cos z, \sin z\;$?

So the basis of the corresponding subspace must be: ... ?.

And the corresponding subspace then has dimension = ... ?

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